Problem 56 - Elementary Algebra

Question

Solve. $$ (x-2) 2-36=0 $$

Step-by-Step Solution

Answer

The solution is \(x = 20\).

Step 1: Simplify the Equation

Start by simplifying the equation. Distribute the multiplication: \((x-2)2 = 2(x-2) = 2x - 4\). Substitute this into the equation: \[ 2x - 4 - 36 = 0 \].

Step 2: Combine Like Terms

Combine the constant terms on the left side of the equation:\[ 2x - 4 - 36 = 2x - 40 \]. So the equation becomes \[ 2x - 40 = 0 \].

Step 3: Solve for x

To isolate \(x\), add 40 to both sides of the equation:\[ 2x - 40 + 40 = 0 + 40 \], which simplifies to:\[ 2x = 40 \].

Step 4: Divide Both Sides by 2

Finally, divide both sides of the equation by 2 to solve for \(x\): \[ \frac{2x}{2} = \frac{40}{2} \], leading to: \[ x = 20 \].

Key Concepts

Algebraic Manipulation

In algebra, we often need to simplify expressions to make them easier to work with. This process is known as algebraic manipulation. In our exercise, the expression \((x-2)2\) requires us to use the distributive property. This property lets us multiply each term within the parentheses by a number outside the parentheses. So, in the expression \((x-2)2\), we multiply both \(x\) and \(-2\) by 2:

• The \(x\) becomes \(2x\).
• The \(-2\) becomes \(-4\).

Together, you get \(2x - 4\). This step simplifies the equation and sets the stage for solving it. Learning to manipulate algebraic expressions is key to solving more complex problems.

Solving Equations

Solving equations is like unlocking a mystery. You need to find the value of the variable that makes the equation true. When you simplify and solve equations, you usually do so in several steps, following a logical order. After distributing numbers, collect like terms to make the equation simpler to read and solve.

In our exercise, we combined \(-4\) and \(-36\), resulting in \(-40\). This combination helps you focus on the core variable part of the equation. Now, the equation reads \(2x - 40 = 0\). Solving equations might seem challenging initially, but it's all about keeping balance by doing the same thing on both sides of the equation.

Variable Isolation

Variable isolation is a crucial step when solving equations. It means we want to get the variable all by itself on one side of the equation. This makes it easy to see its value.

In our exercise, we had the equation \(2x - 40 = 0\). The goal was to isolate \(x\). Here's how we did it:

• We added 40 to both sides to start eliminating the constant term. This results in \(2x = 40\).
• Next, divide every term by 2, the coefficient of \(x\), to solve for \(x\). Thus, \(x = 20\).

By isolating the variable, we found that \(x = 20\). Once you master variable isolation, solving equations becomes more straightforward and less intimidating.